Skip to main content

Full text of "Analytical Mechanics"

See other formats


WORK                                         1
successive infinitesimal displacements.   Therefore the wo done in any displacement is given by the integral
W = I  F cos a ds.                                  i
= P Jo'
When the path of the particle is curved the direction of coincides with that of the tangent to the curve. Therefc F cos a is the tangential component of the force. In ott words the tangential component of force does all the wo] Hence
W= f*FTds.                              _(
*/
The normal component does no work because the par|ii is not displaced along it.                                              —
Special Cases. Case I.—When the force is constant, jbo in direction and in magnitude, it can be taken out of-t integrand. Therefore
W = F I  cos a ds.
<-   Jo
The last integral equals the projection of the path upon t direction of the force. Therefore the product of the foi by the projection of the path upon the line of action of t force equals the work done.
Case II. — When the force is constant and the path straight then the angle between the force and the displa< ment is constant. Therefore
W = F cos a I ds
Jo
= Fs cos a.
Case III.—When the force is not only constant but also parallel to the path, then a = 0. Therefore
W = Fs. Case IV. —When the force is at right angles to the d
placement a = ^, and cos a = 0.    Hence W = 0.    Therefc 2
the force does no work unless it has a component along t